Keep your party small and your characters simple.Characters in 4e can be very complex and it is very tedious to run more than two at a time. I suggest playing only one character at a time. If you absolutely have to (mostly to allow for more tactical options during battle) create two characters, I strongly suggest to stick to Essentials builds. In my opinion, Essentials characters are somewhat streamlined without being "dumbed down" and still provide fun options. Now you might argue that 4e is a game geared towards and balanced for a party of 4 to 5 heroes but that is only the case in published adventures when it comes to combat encounters. Despite it's appearance, 4eis more than a glorified combat engine. Especially when playing solo you can have as much or as little combat as you like and focus on other aspects like interaction or exploration. But even with only one or two characters, you can create fun combat encounters (more on that later).

Use a generic erasable map and generic tokens.Let's face it, you probably would not play 4th Edition if you didn't like tactical gridded combat with miniatures. And that is the big problem when going solo: Since you have no GM and no adventure prepared you don't know what is going to happen so you don't know which minis to get out and which maps you need. And that probably is the problem for many solo gamers trying 4e solo: The flow of the game is disrupted as you have to search for minis, set up the map etc. But how to work around this issue if there are so many beautiful maps around?

This might sound strange at first but there is a simple solution: Don't use them.

By now you probably go like "What the heck is this guy talking about?!" but hear me out. I think what some consider a major weakness of 4e, the dependence on minis, tokens and battle maps, is also it's greatest strength. Because in my opinion, with 4e you actually get two games for the price of one. You get a well balanced role-playing system and also a terrific dungeon crawl board game - and there is the right time for everything.

If you play 4e with friends, in a regular campaign, with a DM who prepares an adventure, then by all means use all your fancy miniatures and colorful poster maps. The DM probably has all the maps prepared anyways in his folder for the session and all his minis or colorful monster tokens sorted out. If you play 4e as a pure dungeon crawl board game, with prepared random encounter tables, like hinted at in products like Dungeon Delve, then you will probably have a very clear idea about where the action is going to take place and which "theme" the quest has. In that case it is easy to prepare your minis and maps in advance and use all the nice accessories like poster maps, dungeon tiles etc. to play a dungeon crawl board game.

A solo session is a very different experience. You work your imagination and get involved in a story which unfolds before you and you don't know what will happen next. Here you just need a way to quickly play out the tactical combats and be prepared for every eventuality. So when I go solo, I only use one miniature for my PC. Other than that I have three wet-erase markers, black, green and blue and a plastic zip bag full of differently colored wooden generic game pieces, as well as a wet-erase battlemat in my portable DM-Kit. This gives me the flexibility to quickly create every encounter area and play out every encounter without spending much time to set things up. You will be surprised how a few green bushes, a blue stream or pond and some black rocks turn into a nice wilderness encounter location!